ON THE MOTION OF PENDULUMS. 53 member of (105) go on decreasing, so that greater and greater accuracy is required in the calculation of the functions MQ9 &c., and of the products LMQ, &c., in order to ensure a given degree of accuracy in the result. The calculation by the descending series (113) is on the contrary very easy. It will be found that the first differences of ttt2&' and of W12(& — 1) are nearly constant, except near the very beginning of the table. Hence in the earlier part of the table the value of Jc or Jc for a value of m not found in the table will be best got by finding m2k — ttt2 or m2&' by interpolation, and thence passing to the value of k or k'. Very near the beginning of the table, interpolation would not succeed, but in such a case recourse may be had to the formulae (103), (104), (105), the calculation of which is comparatively easy when in is small. It did not seem worth while to extend the table beyond m = 4, because where til is greater than 4, the series (113) are so rapidly convergent that k and K may be calculated to a sufficient degree of accuracy with extreme facility. 38. Let us now examine the progress of the functions k and k1. When Itt is very small, we may neglect the powers of m in the numerator and denominator of the fraction in the right-hand member of equation (105), retaining only the logarithms and the constant terms. We thus get L - whence L being given by (102) and (104), or (104) and (10G). When m vanishes, L, which involves the logarithm of in"1, becomes infinite, but ultimately increases more slowly than if it varied as m affected with any negative index however small. Hence it appears from (115), that k — 1 and k' are expressed by m""2 multiplied by two functions of m which, though they ultimately vanish with m, decrease very slowly, so that a considerable change in m makes but a small change in these functions. Now when the radius a of the cylinder varies, everything else remaining the same, m varies as a,